Question

You will use a subset of the mtcars data. Run the following R codes and use the mtcars2 dataset to answer the questions (a)-(d).

R codes:
sct <- sced(50)
idx <- sample(32:25, replace = FALSE)
tcars2 <- mtcars[idx, ]
tcars2Scy <- factor(mtcars2$scyl)
Now the mtcars2 data set contains 25 observations. Consider a regression model where the response is mpg and the two predictors are weight and cylinder. Note that this time we treat the cylinder predictor as a categorical variable with three categories (4, 6, 8). Use the MLR model: Y = β0 + β1*weight + β2*cyl6*weight + β3*cyl8*weight, where β1 is weight, β2 is 1 if cyl=6 and 0 otherwise, and β3 is 1 if cyl=8 and 0 otherwise.

(a) Obtain the fitted value of mpg at weight = 3, cylinder = 8.
(b) At α = 0.05, test that H0: βcyl6 = βcyl8 = 0 vs Ha: At least one of βcyl6 and βcyl8 is nonzero.
Suppose we wonder if there is a significant interaction between the weight and cylinder predictors. Now consider a larger model Y = β0 + β1*weight + β2*cyl6*weight + β3*cyl8*weight + β4*weight*cyl6 + β5*weight*cyl8. Answer (c) and (d) using this model.
(c) Obtain the fitted value of mpg at weight = 3, cylinder = 8.
(d) At α = 0.05, test that H0: βweight:cyl6 = βweight:cyl8 = 0 vs Ha: At least one of βweight:cyl6 and βweight:cyl8 is nonzero.

Answer 1

The provided regression model is used to calculate the fitted value of mpg for a specific weight and cylinder condition. For hypothesis testing of coefficients βcyl6 and βcyl8, a statistical test is used, and the p-value is compared to the significance level of 0.05.

Step-by-step explanation:

As the student has provided R code and is working with a regression model, we can assume they are using statistical methods to analyze a subset of the mtcars data set. The tasks involve calculating fitted values using specified regression models and conducting hypothesis tests regarding the coefficients of the model.

The regression model provided is Y = β0 + β1*weight + β2*cyl6*weight + β3*cyl8*weight, where cylinder status (cyl6 or cyl8) is treated as 1 if the condition is true and 0 otherwise. The fitted value of mpg at a weight of 3, with a cylinder value of 8, would be calculated by substituting these values and the estimated coefficients into the model.

To test the hypothesis H0: βcyl6 = βcyl8 = 0 against the alternative that at least one of them is nonzero, the student would need to run a statistical test such as an ANOVA or t-test and compare the p-value to the significance level of α = 0.05.